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Diffstat (limited to 'source/know/concept/dirac-delta-function/index.md')
| -rw-r--r-- | source/know/concept/dirac-delta-function/index.md | 20 |
1 files changed, 6 insertions, 14 deletions
diff --git a/source/know/concept/dirac-delta-function/index.md b/source/know/concept/dirac-delta-function/index.md index 518eba1..0185b78 100644 --- a/source/know/concept/dirac-delta-function/index.md +++ b/source/know/concept/dirac-delta-function/index.md @@ -65,11 +65,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-scale"/> -<label for="proof-scale">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-scale">Proof.</label> + +{% include proof/start.html id="proof-scale" -%} Because it is symmetric, $$\delta(s x) = \delta(|s| x)$$. Then by substituting $$\sigma = |s| x$$: @@ -77,9 +74,8 @@ $$\begin{aligned} \int \delta(|s| x) \dd{x} &= \frac{1}{|s|} \int \delta(\sigma) \dd{\sigma} = \frac{1}{|s|} \end{aligned}$$ +{% include proof/end.html id="proof-scale" %} -</div> -</div> An even more impressive property is the behaviour of the derivative of $$\delta(x)$$: @@ -89,11 +85,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-dv1"/> -<label for="proof-dv1">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-dv1">Proof.</label> + +{% include proof/start.html id="proof-dv1" -%} Note which variable is used for the differentiation, and that $$\delta'(x - \xi) = - \delta'(\xi - x)$$: @@ -102,9 +95,8 @@ $$\begin{aligned} &= \dv{}{x}\int f(\xi) \: \delta(x - \xi) \dd{x} = f'(x) \end{aligned}$$ +{% include proof/end.html id="proof-dv1" %} -</div> -</div> This property also generalizes nicely for the higher-order derivatives: |